Assignment Brief and Guidance: Scenario You are employed as an Assistant Engineer by a large Design and Build construction firm in their central Birmingham office. The firm has recently appointed a new Managing Director who has given you several tasks for mathematical analysis and solutions. The tasks foc

Assignment Brief and Guidance:
Scenario
You are employed as an Assistant Engineer by a large Design and Build construction firm in their central Birmingham office. The firm has recently appointed a new Managing Director who has given you several tasks for mathematical analysis and solutions.
The tasks focus on a variety of problems and some of them focus on the application in building/civil engineering.
The Managing Director suggests that your solutions covers:

• The methods used to solve the different tasks
• All answers have appropriate units
• All solutions are detailed
• The answers are shown to a reasonable level of accuracy. Clear diagrams are used wherever appropriate

Produce detailed solutions for the following:
1) The most commonly used field surveying instrument is the total station used to measure angles and distances.
This instrument was used by a surveyor to measure vertical angles from the horizontal at stations A & B to the top of an inaccessible chimney.
Calculate the height of the chimney from ground level if angle θ0= 400 20’ 10’’,  angle βo= 520 25’20’’, and the distance between station A & B was measured at 12m.

2) A Total station was used to measure a horizontal angle U, a vertical angle V and a distance R as shown in Figure 2

Calculate the Cartesian coordinates (X, Y, Z) of station P from the polar system (R, U, V) if R is measured at 70.350m, U =70.250 degrees and V = 68.230 degrees
3) A tunnel entrance makes a parabolic arch that can be represented by the quadratic function y = -x² -4x+12 where y is the height of the tunnel and x is the distance across the width of the tunnel in meters. Calculate the height and width of the tunnel.
 
4) A system of linear equations is used to analyse the flow of traffic for a single road intersection.
The system of equations for the intersection was formulated as :
3x + 5y =7
4x – 3y = 19
Calculate the traffic flow rates x & y using ;

1. algebra,
2. graphical method
3. Inverse Matrix.

5) Tension member forces T1 & T2 in a roof truss joint are given by the following system of equations;
5T1 + 7T2 = 9
3T1 – 2T2 =21
Use inverse matrix calculate the tension forces T1 and T2
6) The crushing strengths (unit: N/mm²) of 60 concrete cubes were determined and their results are given here:
Class interval                                 Frequency
32– 34                                                 5
35-37                                                   9
38-40                                                  15
41-43                                                  16
44-46                                                  11
47–49                                                  4

1. Find the mean crushing strength
2. Produce an Ogive and find the median strength
3. Calculate the standard deviation.

7) A machine produces components that are used in bridge constructions.
The mean diameter of the components is 23 mm and the standard deviation is 0.3 mm.
Assuming a normal distribution find the probability that the diameter of the components is between 22.7 mm and 23.2 mm.
8) On a Civil Engineering project 5% of the total concrete cubes tested are shown to below acceptable standard.
If a random sample of 6 cubes is chosen, assuming a Binominal distribution find the probability of getting ;
(a) All 6 samples to be above standard (b) only one below standard.
9) A manufacturer of sprinkler systems designed for fire protection claims that the average activating temperature is at least 134◦F. To test this claim, you randomly select a sample of 30 systems and find the mean activation temperature to be 132◦F with a standard deviation of 3.2◦F.
At α = 0.10, do you have enough evidence to reject the manufacturer’s claim
10) Your manager has asked you to produce a short report in which you are to evaluate the analytical and statistical findings from the tasks completed and justify the techniques and software used to solve the problems.
11) Differentiate the following functions:-

1. y = 5x⁶ – 3x⁶ + 9 with respect to x
2. I = 8sin θ  + cos 8θ with respect to θ
3. y =5 cos(5θ -4) with respect to θ
4. y = ln (2x + sin x) with respect to x
5. y = 8 e ⁽3x+2) with respect to x

12)a). Use integral calculus to determine indefinite integrals of the following functions::-

1. ∫ ( 7x⁴ + 12x – 4 ) dx
2. ∫(sin 3θ – cos 4θ) dθ
3. ∫ (1/ x +  e^5x ) dx
4. ∫ ( e^3x  sin x) dx

12 b). Use integral calculus to determine definite integrals of the following functions;

1. ∫ (3x² + 8x – 2 )  dx  between the limits x = 3 and x = 1
2. ∫ ( sin 3θ – cos 2θ )  dθ between the limits θ = 3 radians and θ = 0 radians
3. ∫(1/ x +  e^2x)dx between the limits x = 4 and x = 1

13 a). The area enclosed by a Shear Force graph gives us the Bending Moment at a particular point.
If the equation of a Shear Force graph is y = -2x + 10
where y represents the Shear Force and x is the distance along the beam.
Use integration to calculate the Bending Moment 1.6 m from the end of the beam
b). A manufacturer makes a metal cone to hold concrete. The volume enclosed by the cone is determined by rotating the line y = 2x + 2.4 around the x-axis between x = 0 and x = 10. Calculate the volume of concrete that it can hold.

14 a) An open rectangular water tank with square ends is to be made from a thin sheet of metal.

• What is the least area of metal for which the volume is 14m3?
• Use the second-order derivative to show that this is a minimum

b) The height of a road surface H (in meters) above a datum is given by the equation:- H = – 0.1 x2 + 0.48x –0.06; Where ‘x’ is the distance from the datum.
(a) Calculate the maximum height of the road above the datum.
(b) Determine the second-order derivative and use it to show that the turning point is maximum.
15) Find the dimensions of the following quantities in terms of M, L and, T:
a)  Kineticenergy  b)acceleration c)Velocity
Use the method of dimensions to check whether the following formulae are dimensionally correct; u and v are velocities, a is acceleration, s is distance and t is time:
a)  u2 = v2– 2as
b)  u = v –at2
c)  v = ½at2
16) A pilling company bores a circular pile 80m deep. Estimate the cost of piling if the cost is £30 for the first meter with an increase of cost of £2 per meter for each succeeding meter.
17) In a standard penetration test (SPT) on sandy clay, the sampler penetrated by 20.4mm mm into the soil due to the second blow of the hammer. If the sampler penetrated by 3.45 mm due to the 11^th blow of the hammer, calculate:

1. the penetration of the sampler due to the 7^th blow of the hammer
2. total penetration of the sampler into the soil.

18) Using Vector analysis techniques solve the following construction-related problems and evaluate the effectiveness and relevance of vector analysis.
a) Determine the magnitude and direction of the resultant of the two forces acting on the lifting hook.

b) A member joint in a roof truss is acted upon by four forces as shown below. Use Vector analysis to determine the magnitude and direction of the resultant of the four forces.